Vibrating Quartz Fork—A Tool for Cryogenic HeliumResearch
M. Blažková · M. Clovecko · V.B. Eltsov · E. Gažo · R. de Graaf · J.J. Hosio ·M. Krusius · D. Schmoranzer · W. Schoepe · L. Skrbek · P. Skyba ·R.E. Solntsev · W.F. Vinen
Abstract Oscillating objects such as discs, piles of discs, spheres, grids and wireshave been widely used in cryogenic fluid dynamics and in quantum fluids researchsince the discovery of superfluidity. A new addition are quartz tuning forks, commer-cially available frequency standards. We review their use as thermometers, pressure-and viscometers as well as their potential as generators and detectors of cavitationand turbulence in viscous and superfluid He liquids.
Keywords Low temperature instrumentation and new techniques · Normal 3He ·Superfluid 3He · Superfluid 4He · Turbulence
PACS 07.10.-h · 47.37.+q · 67.57.De
M. BlažkováInstitute of Physics ASCR, v.v.i., Na Slovance 2, 182 21 Prague, Czech Republic
M. Clovecko · E. Gažo · P. SkybaCenter of Low Temperature Physics, Institute of Experimental Physics, Watsonova 47, 04 001Kosice, Slovakia
V.B. Eltsov · R. de Graaf · J.J. Hosio · M. Krusius · R.E. SolntsevLow Temperature Laboratory, Helsinki University of Technology, 02015-TKK, Helsinki, Finland
D. Schmoranzer · L. Skrbek (�)Faculty of Mathematics and Physics, Charles University, V Holešovickách 2, 180 00 Prague, CzechRepublice-mail: [email protected]
W. SchoepeFakultät für Physik, Universität Regensburg, 93040 Regensburg, Germany
W.F. VinenSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
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1 Introduction
Various types of oscillating structures have been important for probing the hydro-dynamic properties of quantum fluids since the discovery of superfluidity. The An-dronikashvili experiment [1–3], the basis for the two fluid model, was a measurementof the torsional oscillations of a pile of closely spaced discs and was the first directdetermination of the densities of the normal (ρn) and superfluid (ρs) fractions in He II,while measurements with oscillating discs, spheres, capillary flow, or U-tube oscil-lations have been employed to display various “critical velocities” above which theideal two-fluid picture did not hold. The reason for using various oscillating structuresis obvious—in a cryostat it is technically simpler to generate flow with oscillating ob-jects rather than by some more complicated external drive, as would be needed, e.g.,for pipe flow.
In more recent times the use of vibrating wire loops has regained importance.Originally a fine conducting wire stretched along the axis of a cylindrical rotatingcontainer was used by Vinen [4] as a sensor to prove quantization of vorticity inHe II. Today fine superconducting vibrating wires, usually in the form of a half-loopa few mm in diameter, are the most widely used resonators. They have been used forinvestigating the nucleation of quantized vorticity in He II (e.g. [5–7]) and in 3He-B(e.g. [8, 9]), as thermometers, pressure- and viscometers in 4He, 3He and 4He-3Hemixtures over a wide temperature range, even down to microkelvin temperatures [10].They are driven by the Lorentz force from a small ac current bias in an applied mag-netic field.
The quartz tuning fork is a piezoelectric resonator (Fig. 1), which is mass producedas frequency standard for digital watches and which now has found a large numberof other applications [11, 12]. It is cheap, sensitive, robust, and easy to install and touse. Only two wires are needed to drive and readout a fork as sensor; no magneticfield is needed. In fact, the fork is quite insensitive to magnetic fields and becomesthus useful as a field-independent sensor in measurements as a function of magneticfield. A relatively simple electronic scheme composed of a digital oscillator and alock-in amplifier (see, e.g. Ref. [13]) is used for detecting its resonant response overseven orders of magnitude of the driving force in both viscous and quantum fluids.Because of these many convenient features the fork is becoming an indispensablemonitoring tool in the He sample cell, starting from sensitive monitoring of pressuresduring flushing, evacuation, and filling of complicated cell structures, and continuingall the way to liquid 3He thermometry at sub-millikelvin temperatures in the ballisticregime of quasiparticle excitations. This has provided a strong motive to explore theapplicability of the quartz tuning fork further, especially as a generator and sensor ofvortices in the He superfluids.
The quartz tuning fork comes inside a vacuum-tight metal can and when cooled, itsresonant frequency f0 decreases and the Q value increases. At LHe temperatures thereduction in resonant frequency from the room-temperature value (typically 215 Hz)is about 70 Hz and the Q value approaches 106. If the can is removed, the bare fork invacuum behaves in a similar manner: The reduction in the resonant frequency remainsthe same while the Q value is typically lower than for the fork inside its original can.However, the Q value varies from one fork to another, typically in the range from
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Fig. 1 (Color online) Photographs of quartz tuning forks, both with the fork in its original can and withthe can removed (left). The micrographs show the ends of its prongs (middle) and the characteristic surfaceroughness (right)
2 · 104 to 5 · 105. To maintain high Q values, care should be used not to strain thequartz while cutting away the can, when bending the leads, or while replacing themwith non-magnetic wires in magnetic applications. In this short review we brieflysummarize earlier work on their use in He experiments [13–16] and describe somenew results, which further strengthen their potential as a multi-purpose cryogenicsensor.
2 Tuning Fork as Sensor in Linear Regime
At low amplitude in the linear drive regime the quartz tuning fork is used as sec-ondary thermometer, pressure gauge, viscometer, or flowmeter in measurements onnormal and superfluid 4He and 3He. Its advantage is that with a single sensor in oneand the same setup all these different types of tasks can be performed from roomtemperature down to the lowest temperature. In the linear regime the flow aroundthe fork is laminar1 and its response behaves in good accordance with hydrodynamicmodels [13]. Different tuning forks, even if they originate from the same manufac-turing batch, do not appear to obey exactly the same calibration in LHe temperaturemeasurements [13]. However, by preselecting the forks according to their Q values atsome temperature in the LHe range in vacuum, similar behavior can be expected.
In superfluid 3He work, for instance, the fork is a most useful secondary ther-mometer. It requires less work and know how to implement and to operate thanany other of the current methods for thermometry. It can be used as thermome-ter from the normal state down to the very lowest temperatures with good sensi-tivity and seems to provide reproducible readings even in the anisotropic A phase.At temperatures below 0.3Tc its temperature dependence can be extrapolated usingQ = f0/�f ∝ exp (�/kBT ), where �(T ,P ) is the superfluid energy gap. The pro-portionality factor in this calibration can be fixed at one known temperature.
1It appears that the response of the fork depends weakly on the geometry of the surrounding container.This effect is attributed to the existence of steady secondary flow through the action of viscosity in theboundary layer, called “streaming” [17]. As far as we know, this effect has not been taken into account inthe analysis of measurements recorded with oscillating objects, most notably with vibrating wires. It mightbecome important in accurate measurements of physical quantities spanning a large range of values.
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In principle, also in He I and He II both the resonance frequency and width canbe used for thermometry. Owing to the extremely low kinematic viscosity of He Iand of the normal component in He II the temperature-induced changes in the re-sponse of the fork are small. External interference from other sources, such as flowor solid particles raining on the fork in technical helium, may then become a problem.However, the fact that the fork is sensitive to such features makes it useful for otherpurposes, as we will see below, but limits its use as thermometer or pressure gaugein technical 4He in an open dewar. For accurate and reproducible readings the forkshould be housed in a shielded volume or be used in its original can which is onlypartly removed. When filling a separate sample cell where a tuning fork is located,the 4He charge should be introduced via a cryotrap and/or a sufficiently fine filter, toavoid particles of air or water, which would float in the liquid and settle on the quartzsurface.
3 Tuning Fork as Generator and Detector of Turbulence in 4He
At large drive the prongs of the tuning fork vibrate at high velocities up to 10 m/s. Thiscan be used for different types of studies, including cavitation in He I and He II alongthe saturated vapor pressure curve and at slightly elevated pressures. Cavitation isobserved as a breakdown of the resonance response at a critical velocity, while slowlysweeping the frequency of the drive across resonance [18]. A second application athigher drives is the measurement of turbulence.
3.1 Transition from Laminar to Turbulent Drag in Normal Viscous Fluids
The damping of the tuning fork has been investigated by the Prague group in gaseous4He at liquid nitrogen temperature at external pressures up to 30 bar and in normalHe I liquid [15]. For measurements on classical fluid dynamics He is a remarkableworking fluid—a playground where the density and kinematic viscosity can be tunedby orders of magnitude in situ, by changing temperature and pressure in the samplecell. Combined with the large dynamic range of the fork (more than seven decadesover which its response to the harmonic drive can be detected), one has an idealsystem for studying the transition from laminar to turbulent drag and the scaling ofthe critical velocity associated with this transition.
From a measurement of the velocity versus the driving force (Fig. 2), it is foundthat in viscous flow the critical velocity for the crossover from laminar to turbulentdrag in the limit U/ω � � � δ scales as Ucr ∝ √
νω over at least two decades ofkinematic viscosity ν [15]. Here U is the peak velocity of the fork, � its characteristicsize, ω its angular frequency, and δ = √
2ν/ω the viscous penetration depth. Thevalidity of this scaling was recently tested further by performing measurements withforks of various sizes and oscillating at different frequencies.2 The scaling can beexplained qualitatively by equating the linear and the turbulent drag forces at Ucr,using the approach described in Ref. [15].
2The data obtained with different forks in liquid and gaseous helium as well as visualizations of dynami-cally similar flow in water at room temperature using the Baker pH technique will be described elsewhere.
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Fig. 2 (Color online) (Left) Transition from laminar to turbulent flow in 4He, measured with a tuningfork in He I and He II at three temperatures. At low drive level (in the laminar regime) the velocity isa linear function of the applied drive, while after the crossover to the turbulent regime the driving forceis proportional to the square of the velocity. The dashed lines indicate the slopes U ∝ F and U ∝ √
F .(Right) In viscous fluid (He I) the drag coefficient Cd plotted versus velocity tends gradually towardsconstant value at high velocities in the turbulent regime, while in He II the superfluid fraction causes asharp transition at a critical velocity US
cr
3.2 Transition from Laminar to Turbulent Drag in He II
The analysis has been recently extended from viscous He I to superfluid He II (a pre-liminary report is Ref. [16]). The left panel of Fig. 2 shows no appreciable qualitativechange in the character of the dependence of the velocity versus driving force whencrossing Tλ. On decreasing the temperature of He II along the saturated vapor pres-sure curve further, however, the crossover from laminar to turbulent flow becomesgradually sharper and the character of the curve above the critical velocity changes.This change is seen more clearly in the right panel of Fig. 2, where the drag coeffi-cient Cd is plotted for three different temperatures.3 Cd is defined from the equation
F = 1
2CdρAU2, (1)
where ρ is the fluid density and A is the projected area of one prong of the fork ona plane normal to the bulk flow. For laminar viscous flow the drag is approximatelyproportional to U , so that Cd ∼ U−1. We see that in classical viscous He I, in thevicinity of Ucr, the measured dependence Cd(U) gradually levels off and Cd acquires
3The data shown for He I at 4.2 K have been measured at elevated pressure, as at saturated vapor pressurethe fork vibrating at high velocity of order 50 cm/s causes cavitation [18], preventing measurements in theturbulent drag regime up to the highest possible velocities. In He II on the saturated vapor pressure curvecavitation occurs at velocities of order 2 m/s far inside the turbulent regime.
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an approximately constant value of order unity. In He II well below Tλ, Cd(U) be-haves differently. It displays the laminar part, where the drag is due to the viscousnormal fluid only. Beyond a sharp minimum, Cd(U) increases again and displays abroad maximum above which it gradually becomes constant as in the classical case.
This behavior of the drag coefficient Cd(U) can be understood in the framework ofthe two-fluid model, assuming that the total drag consists of two independent contri-butions. The normal fraction behaves similar to any classical viscous fluid and causesthe leveling off in Cd(U) at higher velocities (curve at 1.3 K in right panel of Fig. 2).The critical velocity UN
cr can be determined in analogy with Ucr in He I. The super-fluid fraction produces the sharp minimum where the turbulent drag sets in, which isidentified as its critical velocity US
cr. It was found, at least approximately, to be fre-quency independent, while UN
cr ∝ √ω. Consequently, thanks to the high oscillating
frequency of the fork, the two critical velocities appear well separated, which makesthis analysis possible.
The transition from laminar to turbulent drag regime has been experimentally stud-ied with a number of different vibrating structures of various shapes (such as wires[6, 7], spheres [19–21], grids [22], and forks [16]) in a range of temperatures downto the zero-temperature limit. The observed critical velocity is often found to dependon the prehistory of vortices in the sample cell (i.e. on the presence of trapped vor-tices) and in some cases, e.g., for thin vibrating wires [6, 7], it is seen to be frequencydependent. New measurements on the fork response in the low temperature limit arein progress and their detailed theoretical analysis using computer simulations [23] asillustration should clarify these questions further.
4 Tuning Fork and Andreev Reflection in 3He-B
In superfluid 3He-B the quartz tuning fork has been used in different types of mea-surements on quantized vortices. At present the simplest demonstration of vortexgeneration and its detection with a tuning fork is to use the signal from Andreevscattering, the reflection of quasiparticle excitations from the superflow fields in avortex tangle, as has been observed and explained by Fisher et al. for vibrating wireresonators [25]. Andreev reflection has been demonstrated by the Košice group inmeasurements with a single fork where the quasiparticle excitations are most likelyscattered by potential flow around the vibrating legs of the fork. The Helsinki grouphas used a pair of forks, where one acts as generator of the turbulent vortex tangleand the other as detector of the quasiparticle density.
4.1 Andreev Reflection from Potential Flow Fields
The inset in Fig. 3 (left panel) shows the dependence of the damping force versusvelocity of the tuning fork, measured in 3He-B at 245 µK. Up to velocities of about2 mm/s this dependence appears linear at first sight, without a tendency to saturation,as is usual for vibrating wires [8]. However, a detailed view at very low drives showsthat the measured linewidth �f initially decreases with increasing drive, resulting ina deviation (of up to 10 %) of the damping force from the linear dependence.
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Fig. 3 (Color online) (Left) Dependence of the fork width �f on the fork velocity U , measured at 245 µKand 0.5 bar. The inset shows the corresponding force versus velocity dependence. (Right) The normalizedratio force/velocity plotted versus velocity shows the departure from unity to lower values owing to An-dreev reflection, until other contributions start to interfere. Triangles pointing left (green online), downand up (black and blue online) are measured for the fork at 360 µK, 275 µK and 220 µK. The crosses (redonline) denote the corresponding data for a 13 µm diameter vibrating wire at 220 µK
The motion of an object in 3He-B is damped by the scattering of quasiparticleexcitations. A simple model was introduced in Ref. [8], to express the damping forceper unit area
FT = pF〈nVg〉[
1 − exp
(−λpF
kBTU
)], (2)
where Vg is the group velocity of excitations, pF the Fermi momentum, U the velocityof the object, λ a constant characterizing the velocity field around the object, and〈nVg〉 = n(pF)kBT exp(−�/kBT ) represents the quasiparticle flux. Here n(pF) isthe density of states in momentum space. The assumption that pFU � kBT allowsus to expand exp(−pFU/kBT ) as a Taylor series. Taking into account the first threeterms one gets:
FT = n(pF)p2Fλ exp(−�/kBT )
[1 − λpF
2kBTU
]U. (3)
The first term describes the velocity independent damping coefficient γ responsiblefor the linear damping force F lin
T = γU which is the form commonly used in ther-mometry. The second, velocity dependent term β(U) = γ λpFU/2kBT , represents thecontribution from Andreev reflection to damping. Since the total damping coefficientγ −β(U) is related to the linewidth, �f , the linewidth versus velocity plot measuresthe contribution from Andreev reflection to the total damping (Fig. 3, left panel). Atvery low drives �f is nearly constant, but as the velocity increases, the number ofquasiparticles scattered by the Andreev process increases. However, when scattered,they exchange only a tiny amount of their momenta, of the order of (�/EF)pF,where EF is the Fermi energy and, as a result, the damping and the correspondinglinewidth �f decreases. At sufficiently high velocities, �f begins to increase. The
532 J Low Temp Phys (2008) 150: 525–535
Fig. 4 (Color online) (Left) Side view of the two forks in the sample cylinder which is primarily usedfor NMR measurements [28]. (Right) Thermal resistance RT = �T/Q measured in steady state from thetemperature rise �T of the detector fork while the generator fork is heated at the power level Q. The fittedline corresponds to R0 ≈ 11 K/W, when � = 1.96kBTc is used from Ref. [24]
velocity (∼ 1 mm/s) at which this occurs could be called a critical velocity abovewhich the fork’s motions produce pairbreaking and/or generate turbulence, as hasbeen explained to be the case for vibrating wires [8, 9].
4.2 Andreev Reflection from Turbulent Vortex Tangle
On the left in Fig. 4 the sample container setup with two tuning forks is shown.The forks are mounted next to each other on a support plate which blocks someof the open cross section of the quartz sample cylinder which is 6 mm in diame-ter. The end of the quartz tube is located at a distance ∼ 2 mm from the sinteredheat exchanger surface. When the generator is operated at high drive in the ballis-tic temperature regime (kBT � �), its heat flux Q causes a temperature rise �T toappear, which is registered with the detector. This is caused by a thermal resistanceRT = �T/Q = R0 (kBT/�) exp (�/kBT ), which is dominated by the constrictionin the open cross section in the region around the forks. Following the analysis inRef. [27], the coefficient R0 can be written in the form R0 = 2π2
�3/(p2
F�kBAh).Using the value � = 1.96 kBTc, which we take from Ref. [24] and which we usein our temperature calibration of the fork against the NMR signal, then a fit of the(�T , Q) data in Fig. 4 gives Ah ≈ 3 mm2 for the effective cross section of the chan-nel in the region around the forks. Since the fork measures primarily the quasiparticledensity ∝ exp (−�/kBT ), at higher generator drives the direct contribution from thetemperature rise owing to the power consumption Q has to be subtracted first, todistinguish finer details.
Figure 5 describes the measurement of Andreev reflection from the vortex tangleproduced by the generator fork. This is similar to the Lancaster studies of vibrating
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Fig. 5 (Color online) Generation and detection of turbulence in 3He-B using two forks. The generatoris driven at different steady drive levels, to generate a turbulent tangle which extends to the detector. Thedetector is operated at constant low level. The presence of the tangle is observed as a decrease in thedamping �f of the detector (equivalent to an increase of the oscillation amplitude at resonance). (Left)Original time trace of the amplitude of the detector fork as the drive of the generator fork is switchedbetween low and high levels. (Right) The relative change of the resonance width of the detector versus theoscillation amplitude of the generator, after subtracting for the temperature rise from the thermal resistanceRT. (Inset) Top view of the mounting of the two forks in the sample cylinder
wire or vibrating grid generated turbulence [25, 26]. In the left panel the detectorresponse (as measured) is recorded as a function of time when heating pulses arefed to the generator. The detector is driven at low velocity (below 0.5 mm/s), whilethe square pulses to the generator are at relatively high velocity (5.6 mm/s in theleft panel). During the pulse the damping of the detector output decreases, whicheffectively corresponds to cooling. The apparent cooling happens in spite of the factthat heat is simultaneously added to the system by the generator pulse. The Lancasterinterpretation of this bizarre phenomenon is Andreev reflection from the generator-produced vortex tangle which partially surrounds also the detector and screens itfrom the impinging quasiparticle shower. Since the quasiparticle density around thedetector determines its damping �f , the damping decreases when the screening bythe tangle is effective.
The right panel in Fig. 5 shows that with increasing generator velocity the re-duction of �f further increases. Here we plot the relative change in the detectordamping, (�f − �f0)/�f0, where �f is the measured width of the detector’s reso-nance response and �f0(T ) is its temperature-dependent width in the absence of anyscreening. Thus the temperature rise from the thermal resistance has been taken intoaccount. The resulting dependence is similar to that measured with vibrating wiresin Lancaster [25]: With increasing generator velocity more vortices are produced, theline density in the tangle increases and its spatial extent in volume grows. Both ef-fects increase the screening of the detector and help to extend the cooling to highergenerator drives. Qualitatively this result is independent of which fork is used as gen-erator/detector.4 From the Lancaster measurements it is known that with decreasing
4With their roles reversed the slope in the right panel of Fig. 5 becomes ∼ 20 % steeper.
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Fig. 6 (Color online) Bolometric detection of NMR resonance absorption with a tuning fork. Quasiparti-cles created by resonance absorption within the pick-up coil of the setup in Fig. 4 effuse through the orificeand raise the quasiparticle density in the space with the two tuning forks owing to the thermal resistanceRT. (Left) The NMR line shape measured with the pick-up coil. (Right) The resonance width �f of thefork recorded (with the detector fork of Fig. 5) for linear upward and downward moving NMR field sweeps(of 100 s duration). The largest differences are caused by the finite thermal time constant for the recoveryafter a heat pulse
temperature the decreasing slope in Fig. 5 (right panel) becomes steeper, i.e. the ap-parent cooling becomes stronger. At the lowest generator drives a critical velocity oforder ∼ 0.5 mm/s might exist, where the screening and the tangle formation start, andwhich thus would correspond to the critical velocity in the right panel of Fig. 3.
5 Conclusions
The quartz tuning fork is a practical and sensitive multipurpose tool in liquid he-lium research which can be operated over the entire temperature range from roomtemperature down to sub-millikelvin temperatures. As a sensor it is usually operatedin the linear mode, vibrating at low amplitude with a resonant response which is ofLorentzian shape. In this mode it is useful as pressuremeter, viscometer, secondarythermometer, and bolometer (see the example in Fig. 6 from the sub-millikelvinregime). Both the width of the frequency response and the resonant frequency canbe measured and provide information on the fluid density and viscosity [13]. At highdrive in the non-linear regime the fork can be used to study cavitation [18] and as agenerator of turbulence in both classical and quantum fluids. An advantage is that itcan be operated continuously through the superfluid transition, allowing complemen-tary measurements on viscous and superfluid turbulence. By now the fork has beenused for turbulence studies with promising results in 4He, 3He, as well as in 3He-4Hemixtures [29].
Much of the work with the tuning fork so far has been repetition of earlier mea-surements with vibrating wire resonators. It is not yet clear how well the tuning forkcompares with the most sensitive vibrating wire devices or what will be its value as aprimary measuring instrument for detecting vortices and turbulence. The analysis and
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evaluation of such measurements is currently pursued by several research groups. Ifthese studies are successful, it would ultimately be desirable to obtain quartz oscilla-tors of much smaller size than the present tuning forks.
Acknowledgements The authors thank a number of colleagues for discussions and for informationabout their results prior to publication. This work is supported by research plans MS 0021620834, AVOZ10100520, GACR under 202/05/0218, GAUK 7953/2007, APVV 51-016604, VEGA 2/6168/06, CE I-2/2007, Academy of Finland (grants 213496, 211507), EU Transnational Access Programme (RITA-CT-2003-505313), and by the U.S. Steel Košice s.r.o.
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